N=4 SYM: Extended Superspace, Coloumb Phase

1) As far as I understand, the attempt to extend the usual N=1
superspace to a superspace that corresponds to extended (N=2, 4)
supersymmetry meets the difficulty that additional bosonic coordinates
appear, corresponding to central charges. This means we have an
infinite set of auxilary fields in the theory. This complication may in
certain cases be overcomed in the N=2 case but not in the N=4 case. In
fact, we do have the so-called "analytic superspace" which allows
bypassing the problem for N=4 but it only works on-shell.

A) What does it mean to have an "on-shell" formulation of the theory?
That we can describe observables, but we cannot write down a
path-integral? What is the "on-shell" formulation of N=4 SYM in
analytic superspace?

B) Nevertheless, can we write a path-integral for N=4 SYM in extended
superspace (with the infinite set of auxilary fields included)?

2) The quantum potential for the scalar fields in N=4 SYM has
non-renormalization properties due to supersymmetry. In particular,
there is a non-trivial moduli space of vacua (with dimension 6 times
the rank of the gauge group) corresponding to the vanishing of the
quantum potential. In a generic sector the gauge group is broken down
to a maximal Abelian subgroup (the so-called Coloumb phase). All the
fields of the theory are in the adjoint representation. In
hep-th/9908171, p. 47, the conclusion is drawn that in the Coloumb
phase theory is in fact _free_ (since the adjoint representation is
trivial for an Abelian group). However, what about the interactions
associated with massive gauge bosons?

I wrote:
> 2) The quantum potential for the scalar fields in N=4 SYM has
> non-renormalization properties due to supersymmetry. In particular,
> there is a non-trivial moduli space of vacua (with dimension 6 times
> the rank of the gauge group) corresponding to the vanishing of the
> quantum potential. In a generic sector the gauge group is broken down
> to a maximal Abelian subgroup (the so-called Coloumb phase). All the
> fields of the theory are in the adjoint representation. In
> hep-th/9908171, p. 47, the conclusion is drawn that in the Coloumb
> phase theory is in fact _free_ (since the adjoint representation is
> trivial for an Abelian group). However, what about the interactions
> associated with massive gauge bosons?

Maybe the author meant that the theory flows to a free one in the IR.
That would make sense.

I wrote:
> 2) The quantum potential for the scalar fields in N=4 SYM has
> non-renormalization properties due to supersymmetry. In particular,
> there is a non-trivial moduli space of vacua (with dimension 6 times
> the rank of the gauge group) corresponding to the vanishing of the
> quantum potential. In a generic sector the gauge group is broken down
> to a maximal Abelian subgroup (the so-called Coloumb phase). All the
> fields of the theory are in the adjoint representation. In
> hep-th/9908171, p. 47, the conclusion is drawn that in the Coloumb
> phase theory is in fact _free_ (since the adjoint representation is
> trivial for an Abelian group). However, what about the interactions
> associated with massive gauge bosons?

Maybe the author meant that the theory flows to a free one in the IR.
That would make sense.